LMFDB - Newform orbit 735.2.bg.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [735,2,Mod(16,735)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(735, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([0, 0, 20]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("735.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 735 = 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	735.bg (of order $21$, degree $12$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.86900454856$
Analytic rank:	$0$
Dimension:	$108$
Relative dimension:	$9$ over $\Q(\zeta_{21})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $q$-expansion, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 108 q - q^{2} + 9 q^{3} + 11 q^{4} + 9 q^{5} + 2 q^{6} - 13 q^{7} - 6 q^{8} + 9 q^{9}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ \( 108 q - q^{2} + 9 q^{3} + 11 q^{4} + 9 q^{5} + 2 q^{6} - 13 q^{7} - 6 q^{8} + 9 q^{9} - q^{10} - 4 q^{11} + 11 q^{12} + 6 q^{13} - 32 q^{14} - 18 q^{15} - 41 q^{16} + 5 q^{17} + 6 q^{18} + 31 q^{19} - 22 q^{20} + 11 q^{21} + 17 q^{22} + 24 q^{24} + 9 q^{25} + 66 q^{26} - 18 q^{27} + 29 q^{28} + 18 q^{29} + 6 q^{30} + 3 q^{31} - 69 q^{32} + 3 q^{33} - 6 q^{34} + 2 q^{35} - 22 q^{36} - 8 q^{37} - 118 q^{38} + 18 q^{39} - 39 q^{40} + 17 q^{41} - 30 q^{42} - 22 q^{43} - 24 q^{44} + 9 q^{45} + 20 q^{46} - 6 q^{47} + 96 q^{48} + 27 q^{49} - 12 q^{50} - 44 q^{51} - 26 q^{52} + 3 q^{53} - q^{54} + 15 q^{55} - 98 q^{56} + q^{57} + 30 q^{58} + 18 q^{59} + 11 q^{60} + 8 q^{61} - 26 q^{62} + 9 q^{63} - 102 q^{64} - 24 q^{65} + 65 q^{66} - 6 q^{67} + 98 q^{68} + 6 q^{70} + 45 q^{71} - 39 q^{72} - 72 q^{73} + 16 q^{74} + 9 q^{75} + 24 q^{76} + 18 q^{77} + 36 q^{78} + 49 q^{79} - 48 q^{80} + 9 q^{81} - 162 q^{82} - 11 q^{83} - 10 q^{84} + 25 q^{85} - 133 q^{86} - 16 q^{87} + 106 q^{88} - 33 q^{89} + 2 q^{90} + 18 q^{91} + 100 q^{92} + 3 q^{93} - 53 q^{94} - 11 q^{95} + 57 q^{96} - 172 q^{97} - 24 q^{98} - 76 q^{99}+O(q^{100}) \)

108 * q - q^2 + 9 * q^3 + 11 * q^4 + 9 * q^5 + 2 * q^6 - 13 * q^7 - 6 * q^8 + 9 * q^9 - q^10 - 4 * q^11 + 11 * q^12 + 6 * q^13 - 32 * q^14 - 18 * q^15 - 41 * q^16 + 5 * q^17 + 6 * q^18 + 31 * q^19 - 22 * q^20 + 11 * q^21 + 17 * q^22 + 24 * q^24 + 9 * q^25 + 66 * q^26 - 18 * q^27 + 29 * q^28 + 18 * q^29 + 6 * q^30 + 3 * q^31 - 69 * q^32 + 3 * q^33 - 6 * q^34 + 2 * q^35 - 22 * q^36 - 8 * q^37 - 118 * q^38 + 18 * q^39 - 39 * q^40 + 17 * q^41 - 30 * q^42 - 22 * q^43 - 24 * q^44 + 9 * q^45 + 20 * q^46 - 6 * q^47 + 96 * q^48 + 27 * q^49 - 12 * q^50 - 44 * q^51 - 26 * q^52 + 3 * q^53 - q^54 + 15 * q^55 - 98 * q^56 + q^57 + 30 * q^58 + 18 * q^59 + 11 * q^60 + 8 * q^61 - 26 * q^62 + 9 * q^63 - 102 * q^64 - 24 * q^65 + 65 * q^66 - 6 * q^67 + 98 * q^68 + 6 * q^70 + 45 * q^71 - 39 * q^72 - 72 * q^73 + 16 * q^74 + 9 * q^75 + 24 * q^76 + 18 * q^77 + 36 * q^78 + 49 * q^79 - 48 * q^80 + 9 * q^81 - 162 * q^82 - 11 * q^83 - 10 * q^84 + 25 * q^85 - 133 * q^86 - 16 * q^87 + 106 * q^88 - 33 * q^89 + 2 * q^90 + 18 * q^91 + 100 * q^92 + 3 * q^93 - 53 * q^94 - 11 * q^95 + 57 * q^96 - 172 * q^97 - 24 * q^98 - 76 * q^99

Display 100 coefficients 10 coefficients

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label	$ a_{2} $			$ a_{3} $			$ a_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
16.1	−1.00387	−	2.55782i	0.0747301	+	0.997204i	−4.06860	+	3.77511i	0.826239	−	0.563320i	2.47565	−	1.19221i	−2.03893	−	1.68605i	8.78913	+	4.23262i	−0.988831	+	0.149042i	−2.27031	−	1.54787i
16.2	−0.727465	−	1.85355i	0.0747301	+	0.997204i	−1.44034	+	1.33644i	0.826239	−	0.563320i	1.79400	−	0.863947i	2.50806	−	0.842413i	−0.0630456	−	0.0303612i	−0.988831	+	0.149042i	−1.64520	−	1.12168i
16.3	−0.562496	−	1.43322i	0.0747301	+	0.997204i	−0.271603	+	0.252010i	0.826239	−	0.563320i	1.38717	−	0.668027i	−2.05266	−	1.66931i	−2.26039	−	1.08854i	−0.988831	+	0.149042i	−1.27212	−	0.867313i
16.4	−0.283822	−	0.723168i	0.0747301	+	0.997204i	1.02369	−	0.949843i	0.826239	−	0.563320i	0.699935	−	0.337071i	−1.87358	+	1.86807i	−2.37731	−	1.14485i	−0.988831	+	0.149042i	−0.641880	−	0.437626i
16.5	−0.0818625	−	0.208582i	0.0747301	+	0.997204i	1.42930	−	1.32620i	0.826239	−	0.563320i	0.201881	−	0.0972210i	1.66290	−	2.05785i	−0.797390	−	0.384003i	−0.988831	+	0.149042i	−0.185137	−	0.126224i
16.6	0.200226	+	0.510168i	0.0747301	+	0.997204i	1.24592	−	1.15605i	0.826239	−	0.563320i	−0.493779	+	0.237791i	1.25638	−	2.32841i	1.82680	+	0.879741i	−0.988831	+	0.149042i	0.452822	+	0.308729i
16.7	0.414784	+	1.05685i	0.0747301	+	0.997204i	0.521213	−	0.483615i	0.826239	−	0.563320i	−1.02290	+	0.492603i	1.09127	+	2.41021i	2.77310	+	1.33545i	−0.988831	+	0.149042i	0.938056	+	0.639556i
16.8	0.813688	+	2.07324i	0.0747301	+	0.997204i	−2.17014	+	2.01360i	0.826239	−	0.563320i	−2.00664	+	0.966346i	2.21808	+	1.44227i	−1.92722	−	0.928102i	−0.988831	+	0.149042i	1.84020	+	1.25463i
16.9	0.865479	+	2.20521i	0.0747301	+	0.997204i	−2.64777	+	2.45677i	0.826239	−	0.563320i	−2.13436	+	1.02785i	−2.61239	+	0.418813i	−3.44056	−	1.65688i	−0.988831	+	0.149042i	1.95733	+	1.33448i
46.1	−1.00387	+	2.55782i	0.0747301	−	0.997204i	−4.06860	−	3.77511i	0.826239	+	0.563320i	2.47565	+	1.19221i	−2.03893	+	1.68605i	8.78913	−	4.23262i	−0.988831	−	0.149042i	−2.27031	+	1.54787i
46.2	−0.727465	+	1.85355i	0.0747301	−	0.997204i	−1.44034	−	1.33644i	0.826239	+	0.563320i	1.79400	+	0.863947i	2.50806	+	0.842413i	−0.0630456	+	0.0303612i	−0.988831	−	0.149042i	−1.64520	+	1.12168i
46.3	−0.562496	+	1.43322i	0.0747301	−	0.997204i	−0.271603	−	0.252010i	0.826239	+	0.563320i	1.38717	+	0.668027i	−2.05266	+	1.66931i	−2.26039	+	1.08854i	−0.988831	−	0.149042i	−1.27212	+	0.867313i
46.4	−0.283822	+	0.723168i	0.0747301	−	0.997204i	1.02369	+	0.949843i	0.826239	+	0.563320i	0.699935	+	0.337071i	−1.87358	−	1.86807i	−2.37731	+	1.14485i	−0.988831	−	0.149042i	−0.641880	+	0.437626i
46.5	−0.0818625	+	0.208582i	0.0747301	−	0.997204i	1.42930	+	1.32620i	0.826239	+	0.563320i	0.201881	+	0.0972210i	1.66290	+	2.05785i	−0.797390	+	0.384003i	−0.988831	−	0.149042i	−0.185137	+	0.126224i
46.6	0.200226	−	0.510168i	0.0747301	−	0.997204i	1.24592	+	1.15605i	0.826239	+	0.563320i	−0.493779	−	0.237791i	1.25638	+	2.32841i	1.82680	−	0.879741i	−0.988831	−	0.149042i	0.452822	−	0.308729i
46.7	0.414784	−	1.05685i	0.0747301	−	0.997204i	0.521213	+	0.483615i	0.826239	+	0.563320i	−1.02290	−	0.492603i	1.09127	−	2.41021i	2.77310	−	1.33545i	−0.988831	−	0.149042i	0.938056	−	0.639556i
46.8	0.813688	−	2.07324i	0.0747301	−	0.997204i	−2.17014	−	2.01360i	0.826239	+	0.563320i	−2.00664	−	0.966346i	2.21808	−	1.44227i	−1.92722	+	0.928102i	−0.988831	−	0.149042i	1.84020	−	1.25463i
46.9	0.865479	−	2.20521i	0.0747301	−	0.997204i	−2.64777	−	2.45677i	0.826239	+	0.563320i	−2.13436	−	1.02785i	−2.61239	−	0.418813i	−3.44056	+	1.65688i	−0.988831	−	0.149042i	1.95733	−	1.33448i
121.1	−0.198450	+	2.64813i	0.955573	−	0.294755i	−4.99556	−	0.752960i	−0.733052	−	0.680173i	0.590917	+	2.58898i	−2.63520	+	0.236022i	1.80347	−	7.90153i	0.826239	−	0.563320i	1.94666	−	1.80624i
121.2	−0.138019	+	1.84174i	0.955573	−	0.294755i	−1.39529	−	0.210306i	−0.733052	−	0.680173i	0.410975	+	1.80060i	0.0317535	+	2.64556i	−0.242043	+	1.06046i	0.826239	−	0.563320i	1.35388	−	1.25621i

See next 80 embeddings (of 108 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
49.g	even	21	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.bg.a	✓	108
49.g	even	21	1	inner	735.2.bg.a	✓	108

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.bg.a	✓	108	1.a	even	1	1	trivial
735.2.bg.a	✓	108	49.g	even	21	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{108} + T_{2}^{107} - 14 T_{2}^{106} - 15 T_{2}^{105} + 115 T_{2}^{104} + 210 T_{2}^{103} + \cdots + 568393281 $ T2^108 + T2^107 - 14*T2^106 - 15*T2^105 + 115*T2^104 + 210*T2^103 - 319*T2^102 - 2074*T2^101 - 2805*T2^100 + 13540*T2^99 + 36487*T2^98 - 31422*T2^97 - 145976*T2^96 - 242138*T2^95 + 278828*T2^94 + 2879197*T2^93 - 1405112*T2^92 - 7907709*T2^91 + 46615510*T2^90 - 28687011*T2^89 - 522686737*T2^88 + 172944308*T2^87 + 2498200340*T2^86 + 179612407*T2^85 - 2274670512*T2^84 - 5384608925*T2^83 - 41790243502*T2^82 + 10711470822*T2^81 + 321952431752*T2^80 + 388710415901*T2^79 - 185628620978*T2^78 - 1501002300151*T2^77 - 4294982055884*T2^76 - 3351520239955*T2^75 + 7001039909015*T2^74 + 31402124481896*T2^73 + 91156482467127*T2^72 - 5458265905358*T2^71 - 571696547796709*T2^70 - 946605053367809*T2^69 + 254898063988400*T2^68 + 4420799118999573*T2^67 + 10737967037806696*T2^66 + 432732958651464*T2^65 - 43783454245880730*T2^64 - 57085537636597044*T2^63 + 56052951155480069*T2^62 + 266042258847372804*T2^61 + 343490471055082290*T2^60 - 281474598581361373*T2^59 - 1608738225266807398*T2^58 - 1322701489561756579*T2^57 + 2699085926935869406*T2^56 + 7199484537976473897*T2^55 + 4170715748086230254*T2^54 - 11709015073062351565*T2^53 - 26092786215162783640*T2^52 - 5062066033188815538*T2^51 + 53245163689542393739*T2^50 + 85699457430662138192*T2^49 + 16885408183704126428*T2^48 - 132132177920401613259*T2^47 - 172405784400909547178*T2^46 + 5337434493587259444*T2^45 + 126791715245236401012*T2^44 + 47733828656760648435*T2^43 + 137988643476562333776*T2^42 + 251749053256160348638*T2^41 - 173875951906274405752*T2^40 - 492729865052280752403*T2^39 - 36123554423619018481*T2^38 + 263943835891947894731*T2^37 + 106367127918647319942*T2^36 + 195456116460661560849*T2^35 - 3311450294782207253*T2^34 - 542541208102048455938*T2^33 - 186028663058163257777*T2^32 + 476066680777444952820*T2^31 + 317964308256568184006*T2^30 + 8427366367863111072*T2^29 - 149322019057063359822*T2^28 - 310466277035706594604*T2^27 - 93192270072286062117*T2^26 + 156982209626862117148*T2^25 + 83865710725740155025*T2^24 + 53101837640060732637*T2^23 + 6051062450921694588*T2^22 - 69713968084105130147*T2^21 + 2326043546678124131*T2^20 + 192888926741743522*T2^19 - 8975430732776205992*T2^18 + 18126347175749855384*T2^17 - 5241020714199346496*T2^16 - 1057073711169006363*T2^15 + 3162516534710247085*T2^14 - 2607241169180747007*T2^13 + 624453215513065093*T2^12 - 103702438950326718*T2^11 + 45921932754854922*T2^10 - 656914130230365*T2^9 + 9597550423283586*T2^8 - 3636248831200485*T2^7 + 459452931332013*T2^6 + 49375348232481*T2^5 + 85519503813861*T2^4 + 9809820761166*T2^3 + 610789422474*T2^2 - 41998663215*T2 + 568393281 acting on $S_{2}^{\mathrm{new}}(735, [\chi])$.

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.bg (of order \(21\), degree \(12\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 16.9
Significant digits:
Format:

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